Mathematical Physics

One way of constructing mirror partners to Fano varieties is via toric degenerations. The case in which this is best understood is the Grassmannian, using the well-known SAGBI basis of the Plucker coordinate ring indexed by semi-standard Young tableaux (SSYT). The mirror construction goes back to work of Eguchi—Hori—Xiong, however its geometry and combinatorics still plays an important role in current mirror constructions. In this talk, I will give an overview of this story, then turn to the question of what can be generalized for Kronecker moduli spaces. Like Grassmannians (which they generalize), Kronecker moduli spaces are high Fano index Picard rank 1 smooth Fano varieties. I will introduce linked SSYT pairs, which play the analogous role of SSYT for Grassmannians in understanding the coordinate ring of the Kronecker moduli space. This is joint work with Liana Heuberger.

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After briefly reviewing the dictionary between branes of Type IIB string theory and representations of the simplest quantum toroidal algebra, I will present a new class of brane setups which I call spiralling branes. Partition functions of these remarkable configurations reproduce K-theoretic vertex partition function, while on the algebraic side they correspond to Drinfeld twists of R-matrices. Time permitting, I will also discuss other applications of spiralling branes.

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The prototypical example of a symmetric tensor category is Rep(G) for G a compact group. The other main source of symmetric tensor categories are Deligne's interpolation categories (S_t, GL_t, and O_t) which extend a family of group representation categories defined at positive integers to all complex numbers. Harman and Snowden have developed a theory of symmetric tensor categories coming from Oligomorphic permutation groups together with a well-behaved measure on G-sets. This includes Deligne's S_t which comes from the infinite symmetric group together with a measure where the usual permutation G-set has volume t. The simplest new example of their theory is the group of order-preserving bijections of the real line with the measure given by Euler characteristic. In joint work with Harman and Snowden (arXiv:2211.15392) we give a detailed description of this new symmetric tensor category, which has a number of novel properties. We call this category the Delannoy category because the dimensions of Hom spaces are given by Delannoy numbers. In this talk I'll outline our main results, including the classification of simple objects, the tensor product rules, and a combinatorial model for the category using Delannoy paths.

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